Equiconsistencies at subcompact cardinals (English)

In: Archive for mathematical logic   ;  55 ,  1  ;  2016
  • ISSN:
  • Article (Journal)  /  Print

How to get this document?

We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □ δ fail, then δ is subcompact in a class inner model. If in addition □(δ +) fails, we prove that δ is $${\Pi_1^2}$$ Π 1 2 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a $${\Pi_1^2}$$ Π 1 2 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n <� ω. We state some results at this level, and indicate how they are proved.

Table of contents – Volume 55, Issue 1

Show all volumes and issues

Tables of content are generated automatically and are based on records of articles contained that are available in the TIB-Portal index. Due to missing records of articles, the volume display may be incomplete, even though the whole journal is available at TIB.

1
Dedicated to Richard Laver
| 2015
3
Indestructibility and destructible measurable cardinals
Apter, Arthur W. | 2015
19
Superstrong and other large cardinals are never Laver indestructible
Bagaria, Joan / Hamkins, Joel David / Tsaprounis, Konstantinos / Usuba, Toshimichi | 2015
37
Generalizing Kruskal’s theorem to pairs of cohabitating trees
Carlson, Timothy | 2015
49
Laver’s results and low-dimensional topology
Dehornoy, Patrick | 2015
85
Baire spaces and infinite games
Galvin, Fred / Scheepers, Marion | 2015
105
Strong measure zero in separable metric spaces and Polish groups
Hrušák, Michael / Wohofsky, Wolfgang / Zindulka, Ondřej | 2016
133
Laver and set theory
Kanamori, Akihiro | 2016
165
A remark on hereditarily nonparadoxical sets
Komjáth, Péter | 2015
177
Left division in the free left distributive algebra on many generators
Miller, Sheila K. | 2015
207
Equiconsistencies at subcompact cardinals
Neeman, Itay / Steel, John | 2015
239
ZF + DC + AX4
Shelah, Saharon | 2016
295
Martin’s maximum revisited
Viale, Matteo | 2015
319
The Weak Ultrafilter Axiom
Woodin, W. Hugh | 2015